Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42664 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,4)(3,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,35,29,34,43,32)(27,36,39,45,38,31,44,47)(28,33,42,51,41,50,54,48)(37,49,52,55,40,46,53,56);(1,2)(3,8)(4,5)(6,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,29,38)(26,31,34,36)(28,40,41,37)(30,39,43,44)(32,45,35,47)(33,49,50,46)(42,53,54,52)(48,56,51,55);(1,3,5,7)(2,6,4,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,43)(31,45)(33,48)(34,35)(36,47)(39,44)(42,54)(46,55)(49,56)(50,51)(52,53);(3,7)(6,8)(11,19)(14,17)(18,24)(21,23)(25,28,29,41)(26,33,34,50)(27,37,38,40)(30,42,43,54)(31,46,36,49)(32,48,35,51)(39,52,44,53)(45,55,47,56);(1,4,5,2)(3,8,7,6)(10,16)(14,17)(15,22)(21,23)(26,34)(27,38)(28,41)(32,35)(39,44)(42,54)(46,49)(55,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7394206);